Collisions of the supercritical Keller-Segel particle system

We consider some scalar parameter $\theta>0$ and a number $N\geq 2$ of particles with positions $X_{t}=(X^{1}_{t},\dots,X^{N}_{t})\in({\mathbb{R}}^{2})^{N}$ at time $t\geq 0$. Informally, we assume that the dynamics of these particles are given by the system of S.D.E.s

where the $2$-dimensional Brownian motions $((B^{i}_{t})_{t\geq 0})_{i\in[\![1,N]\!]}$ are independent. In other words, we have $N$ Brownian particles in the plane interacting through an attraction in $1/r$, which is Coulombian in dimension $2$. Actually, this S.D.E. does not clearly make sense, due to the singularity of the drift, and we will use, as suggested by Cattiaux-Pédèches [4], the theory of Dirichlet spaces, see Fukushima-Oshima-Takeda [11].

This particle system is very natural from a physical point of view, because, as we will see, there is a tight competition between the Brownian excitation and the Coulombian attraction. It can also be seen as an approximation of the famous Keller-Segel equation [16], see also Patlak [20]. This nonlinear P.D.E. has been introduced to model the collective motion of cells, which are attracted by a chemical substance that they emit. It is well-known that a phase transition occurs: if the intensity of the attraction is small, then there exist global solutions, while if the attraction is large, the solution explodes in finite time.

We will show that this phase transition already occurs at the level of the particle system (1): there exist global (very weak) solutions if $\theta\in(0,2)$ (subcritical case, see Proposition 1.3 below), but solutions must explode in finite time if $\theta\geq 2$ (supercritical case).

To our knowledge, the supercritical case has not been studied in details, and we aim to describe precisely the explosion phenomenon. Informally, we will show the following (see Theorem 1.5 below). We assume that $\theta\geq 2$ and $N>3\theta$, we set $k_{0}=\lceil 2N/\theta\rceil\in[\![7,N]\!]$. There exists a (very weak) solution $(X_{t})_{t\in[0,\zeta)}$ to (1), with $\zeta<\infty$ a.s. and such that $X_{\zeta-}=\lim_{t\to\zeta-}X_{t}$ exists. Moreover, there is a cluster containing precisely $k_{0}$ particles in the configuration $X_{\zeta-}$, and no cluster containing strictly more than $k_{0}$ particles. Such a cluster containing $k_{0}$ particles is inseparable, so that (1) is meaningless (even in a very weak sense) after $\zeta$. Just before explosion, there are infinitely many $k_{1}$-ary collisions, where $k_{1}=k_{0}-1$. If $(k_{0}-3)(2-(k_{0}-2)\theta/N)<2$, we set $k_{2}=k_{1}-2$ and just before each $k_{1}$-ary collision, there are infinitely many $k_{2}$-collisions. Else, we set $k_{2}=k_{1}$. In any case, there are infinitely many binary collisions just before each $k_{2}$-ary collision. During the whole time interval $[0,\zeta)$, there are no $k$-ary collisions, for any $k\in[\![3,k_{2}-1]\!]$.

This phenomenon seems surprising and original, in particular because of the gap between binary and $k_{2}$-ary collisions.

We introduce, for all $K\subset[\![1,N]\!]$ and all $x=(x^{1},\dots,x^{N})\in({\mathbb{R}}^{2})^{N}$,

Here $|K|$ is the cardinal of $K$ and $\|\cdot\|$ stands for the Euclidean norm in ${\mathbb{R}}^{2}$. Observe that $R_{K}(x)=0$ if and only if all the particles indexed in $K$ are at the same place. We also set, for $k\geq 2$,

which represents the set of configurations with no cluster of $k$ (or more) particles. Observe that $E_{k}=({\mathbb{R}}^{2})^{N}$ for all $k>N$.

We recall that a squared Bessel process $(Z_{t})_{t\geq 0}$ of dimension $\delta\in{\mathbb{R}}$ is a nonnegative solution, killed when it reaches $0$ if $\delta\leq 0$, of the equation

where $(W_{t})_{t\geq 0}$ is a $1$-dimensional Brownian motion. We then say that $(\sqrt{Z_{t}})_{t\geq 0}$ is a Bessel process of dimension $\delta$. This process has the following property, see Revuz-Yor [21, Chapter XI]: $\bullet$ if $\delta\geq 2$, then a.s., for all $t>0$, $Z_{t}>0$;

Applying informally the Itô formula, one finds that $Y_{t}=\sqrt{Z_{t}}$ should solve

which resembles (1) in that we have a Brownian excitation in competition with an attraction by $0$, or a repulsion by $0$, depending on the value of $\delta$, proportional to $1/r$. This formula rigorously holds true only when $\delta>1$, see [21, Chapter XI].

Consider a (possibly very weak) solution $(X_{t})_{t\geq 0}$ to (1). As we will see, when fixing a subset $K\subset[\![1,N]\!]$ and when neglecting the interactions between the particles indexed in $K$ and the other ones, one finds that the process $(R_{K}(X_{t}))_{t\geq 0}$ behaves like a squared Bessel process with dimension $d_{\theta,N}(|K|)$, where

Similar computations already appear in Haškovec-Schmeiser [12], see also [8]. A little study, see Appendix A, see also Figure 1 and Subsection 1.8 for numerical examples, shows the following facts. For $r\in{\mathbb{R}}_{+}$, we set $\lceil r\rceil=\min\{n\in{\mathbb{N}}:n\geq r\}$.

We thus expect that there may be some non sticky $k$-ary collisions for $k\in\{2,k_{2},k_{1}\}$, some sticky $k$-ary collisions when $k\geq k_{0}$, but no $k$-ary collision for $k\in[\![3,k_{2}-1]\!]$.

As we will see in Subsection 3.13, the S.D.E. (1) cannot have a solution in the classical sense, at least when $d_{\theta,N}(k_{1})\in(0,1)$, because the drift term cannot be integrable in time. We will thus define a solution through the theory of the Dirichlet spaces.

For $x=(x^{1},\dots,x^{N})\in({\mathbb{R}}^{2})^{N}$ and for ${\rm d}x$ the Lebesgue measure on $({\mathbb{R}}^{2})^{N}$, we set

where $\{1\leq i\neq j\leq N\}$ stands for the set $\{(i,j)\in[\![1,N]\!]^{2}:i\neq j\}$. Informally, the generator of the solution to (1) is given by ${\mathcal{L}}^{X}$, where for $\varphi\in C^{2}(({\mathbb{R}}^{2})^{N})$,

see (4) for the last equality. It is well-defined for all $x\in E_{2}$ and $\mu$-symmetric. Indeed, an integration by parts shows that

As we will see in Proposition A.1, the measure $\mu$ is Radon on $({\mathbb{R}}^{2})^{N}$ in the subcritical case $\theta\in(0,2)$, while it is Radon on $E_{k_{0}}$ (and not on $E_{k_{0}+1}$) in the supercritical case $\theta\geq 2$. This will allow us to use some results found in Fukushima-Oshima-Takeda [11] and to obtain the following existence result.

We refer to Subsection B.1 for a quick summary about the notions used in this proposition: diffusion (i.e. continuous Hunt process), link between its generator, semi-group and Dirichlet space, definition of the one-point compactification topology endowing ${\mathcal{X}}_{\triangle}$, etc. Let us mention that by definition, $\triangle$ is absorbing, i.e. $X_{t}=\triangle$ for all $t\geq\zeta$. Also, $t\mapsto X_{t}$ is a priori continuous on $[0,\infty)$ only for the one-point compactification topology on ${\mathcal{X}_{\triangle}}$, which precisely means that it is continuous for the usual topology of $({\mathbb{R}}^{2})^{N}$ during $[0,\zeta)$, and it holds that $\zeta=\lim_{n\to\infty}\inf\{t\geq 0:X_{t}\notin{\mathcal{K}}_{n}\}$ for any increasing sequence of compact subsets $({\mathcal{K}}_{n})_{n\geq 1}$ of $E_{k_{0}}$ such that $\cup_{n\geq 1}{\mathcal{K}}_{n}=E_{k_{0}}$.

As we will see in Remark 11.6, for all $x\in E_{2}$, under ${\mathbb{P}}^{X}_{x}$, $X_{t}$ solves (1) during $[0,\sigma)$, where $\sigma=\inf\{t\geq 0:X_{t}\notin E_{2}\}$. By the Markov property, this implies $X_{t}$ solves (1) during any open time-interval on which it does not visit ${\mathcal{X}}\setminus E_{2}$.

When $\theta<2$, we have $k_{0}>N$ and thus $E_{k_{0}}=({\mathbb{R}}^{2})^{N}$. We will easily prove the following non-explosion result, which is almost contained in Cattiaux-Pédèches [4], who treat the case where $\theta\in(0,2(N-2)/(N-1))$.

When $\theta\geq 2$, we will see that there is explosion. Note that any collision of a set of $k\geq k_{0}$ particles makes the process leave $E_{k_{0}}$ and thus explode. However, it is not clear at all at this point that explosion is due to a precise collision: the process could explode because it tends to infinity (which is not hard to exclude) or to the boundary of $E_{k_{0}}$ with possibly many oscillations.

To avoid any confusion, let us define precisely what we call a collision.

The main result of this paper is the following description of the explosion phenomenon.

The condition $\theta\geq 2$ is crucial to guarantee that $k_{0}\leq N$. On the contrary, we impose $N>3\theta$ for simplicity, because Lemma 1.1 does not hold true without this assumption. The other cases may also be studied, but we believe this is not very restrictive: $N$ is thought as very large when compared to $\theta$, at least as far as the approximation of the Keller-Segel equation is concerned.

Let us mention that the very precise values of $N$ and $\theta$ influence the value $k_{2}$.

(a) If $N=200$ and $\theta=4.04$, we have $k_{0}=100$, $k_{1}=99$ and $k_{2}=98$.

(b) If $N=200$ and $\theta=4.015$, we have $k_{0}=100$ and $k_{1}=k_{2}=99$.

Let us describe informally, in the chronological order, what happens e.g. in case (b) above. We start with $200$ particles at $200$ different places. During the whole story, there is no $k$-ary collision for $k=3,\dots,98$. Here and there, two particles meet, they collide an infinite number of times, but manage to separate. Then at some times, we have $98$ particles close to each other and there are many binary collisions. Then, if a $99$-th particle arrives in the same zone (and this eventually occurs), there are infinitely many $99$-ary collisions, with infinitely many binary collisions of all possible pairs before each. These $99$ particles may manage to separate forever, or for a large time, but if a $100$-th particle arrives in the zone (and this situation eventually occurs), then there are infinitely many $99$-ary collisions of all the possible subsets and, finally, a $100$-ary collision producing explosion, and the story is finished. Informally, the resulting cluster is not able to separate, because the attraction dominates the Brownian excitation, since a Bessel process of dimension $d_{\theta,N}(100)\leq 0$ is absorbed when it reaches $0$. We hope to be able, in a future work, to propose and justify a model describing what happens after explosion.

In many papers about the Keller-Segel equation, the parameter $\chi=4\pi\theta$ is used, so that the transition at $\theta=2$ corresponds to the transition at $\chi=8\pi$. As already mentioned, this nonlinear P.D.E. has been introduced to model the collective motion of cells, which are attracted by a chemical substance that they emit. It describes the density $f_{t}(x)$ of particles (cells) with position $x\in{\mathbb{R}}^{2}$ at time $t\geq 0$ and writes, in the so-called parabolic-elliptic case,

Informally, this solution should be the mean-field limit of the particle system (1) as $N\to\infty$.

We refer to the recent review paper on (7) by Arumugam-Tyagi [1]. The best existence of a global solution to (7), including all the subcritical parameters $\theta\in(0,2)$, is due to Blanchet-Dolbeault-Perthame [2]. The blow-up of solutions to (7), in the supercritical case $\theta>2$, has been studied e.g. by Fatkullin [7] and Velasquez [24, 25]. More close to our study, Suzuki [23] has shown, still in the supercritical case, the appearance of a Dirac mass with a precise (critical) weight, at explosion. This is the equivalent, in the limit $N\to\infty$, to the fact that $\lim_{t\to\zeta-}X_{t}$ exists and corresponds to a $K$-collision, for some $K\subset[\![1,N]\!]$ with precise cardinal $k_{0}$. Let us finally mention Dolbeault-Schmeiser [6], who propose a post-explosion model in the supercritical case. Concerning particle systems associated with (7), let us mention Stevens [22], who studies a physically more complete particle system with two types of particles, for cells and chemo-attractant particles, with a regularized attraction kernel. Haškovec and Schmeiser [12, 13] study a particle system closer to (1), but with, again, a regularized attraction kernel. Cattiaux-Pédèches [4], as well as [8], study the system (1) without regularization in the subcritical case: existence of a global solution to (1) has been shown in [8] when $\theta\in(0,2(N-2)/(N-1))$, and uniqueness of this solution has been established in [4]. Also, the theory of Dirichlet spaces has been used in [4] to build a solution to (1). Finally, the limit as $N\to\infty$ to a solution of (7) is proved in [8] in the very subcritical case where $\theta\in(0,1/2)$, up to extraction of a subsequence. This last result has been improved by Bresch-Jabin-Wang [3], who remove the necessity of extracting a subsequence and consider the (still very subcritical) case where $\theta\in(0,1)$. Olivera-Richard-Tomasevic [18] have recently established the $N\to\infty$ convergence of a smoothed version of (1), for all the subcritical cases $\theta\in(0,2)$. Informally, in view of the mean distance between particles, the regularization used in [18] is not far from being physically reasonable. There is also a related paper of Jabir-Talay-Tomasevic [14] about a one-dimensional but more complicated parabolic-parabolic model.

Let us finally mention the seminal paper of Osada [19], see also [9] for a more recent study, which concerns the vortex model: this is very close to (1), but the attraction $-x/|x|^{2}$ is replaced by a rotating interaction $x^{\perp}/|x|^{2}$, so that particles never encounter.

To our knowledge, this is the first study of the supercritical Keller-Segel particle system near explosion. We hope that this model, which makes compete diffusion and Coulomb interactions, is very natural from a physical point of view, beyond the Keller-Segel community. The phenomenon we discovered seems surprising and original, in particular because of the gap between binary and $k_{2}$-ary collisions. We are not aware of other works, possibly dealing with other models, showing such a behavior.

In Section 3, we give the main arguments of the proofs, with quite a high level of precision, but ignoring the technical issues. While it is rather clear, intuitively, that the process explodes in finite time when $\theta\geq 2$ and that no $K$-collisions may occur for $|K|\in[\![3,k_{2}-1]\!]$, the continuity at explosion is delicate, and some rather deep arguments are required to show that that each $k_{2}$-ary collision is preceded by many binary collisions, that each $k_{1}$-ary collision is preceded by many $k_{2}$-ary collisions, that explosion is preceded by many $k_{1}$-ary collisions, and that explosion is due to the emergence of a cluster with precise size $k_{0}$ (which more or less says that a possible $(k_{0}+1)$-ary collision would necessarily be preceded by a $k_{0}$-collision).

Actually, the rigorous proofs are made technically much more involved than those presented in Section 3, because we have to use the theory of Dirichlet spaces. Due to the singularity of the interactions and to the occurrence of many collisions near explosion, we can unfortunately not, as already mentioned, deal at the rigorous level directly with the S.D.E. (1). We thus have to use some suitable heavy versions of some usual tools such as Itô’s formula, Girsanov’s theorem, time-change, etc.

In Section 2, we introduce some notation of constant use. In Section 3, we explain the main ideas of the proofs, with quite a high level of precision, but without speaking of the heavy technical issues related to the use of the theory of Dirichlet spaces. Section 4 is devoted to the existence of a first version of the Keller-Segel process, namely without the property that ${\mathbb{P}}^{X}_{x}\circ X_{t}^{-1}$ has a density, and we introduce a spherical Keller-Segel process. In Section 5, we show that the Keller-Segel process enjoys a crucial and noticeable decomposition in terms of a $2$-dimensional Brownian motion, a squared Bessel process and a spherical process. Section 6 consists in building some smooth approximations of some indicator functions that behave well under the action of the generator ${\mathcal{L}}^{X}$. In Section 7, we make use of the Girsanov theorem to prove that when two sets of particles of a $KS$-process are not too close from each other, they behave as two independent smaller $KS$-processes. In Section 8, we study explosion and continuity (in the usual sense) at the explosion time. Section 9 is devoted to establish some parts of Theorem 1.5 for some particular ranges of values of $N$ and $\theta$. Using the results of Section 7, we reduce the general study to the special cases of Section 9 and we prove, in Section 10, that the conclusions of Theorem 1.5 hold true quasi-everywhere. Finally, in Section 11, we remove the restriction quasi-everywhere and conclude the proofs of Propositions 1.2 and 1.3 and of Theorem 1.5. Appendix A contains a few elementary computations: proof of Lemma 1.1, proof that $\mu$ is Radon on $E_{k_{0}}$, and study of a similar measure on a sphere. We end the paper with Appendix B, that summarizes all the notions and results about Dirichlet spaces and Hunt processes we shall use.

The existence of the $KS(\theta,N)$-process $(X_{t})_{t\in[0,\zeta)}$, with values in $E_{k_{0}}$, is an easy application of Fukushima-Oshima-Takeda [11, Theorem 7.2.1]. The only difficulty is to show that the invariant measure $\mu$ is a Radon on $E_{k_{0}}$, see Proposition A.1. The process may explode, i.e. get out of any compact subset of $E_{k_{0}}$ in finite time. Observe that a typical compact subset of $E_{k_{0}}$ is of the form, for $\varepsilon>0$,

One can verify, using Itô’s formula, that the center of mass $S_{[\![1,N]\!]}(X)$ is a $2$-dimensional Brownian motion with diffusion constant $N^{-1/2}$, that the dispersion process $R_{[\![1,N]\!]}(X)$ is a squared Bessel process with dimension $d_{\theta,N}(N)$, recall (2), and that these two processes are independent.

Consequently, if $\zeta<\infty$, the limits $\lim_{t\to\zeta-}S_{[\![1,N]\!]}(X_{t})$ and $\lim_{t\to\zeta-}R_{[\![1,N]\!]}(X_{t})$ a.s. exist, and this implies that $\limsup_{t\to\zeta-}||X_{t}||<\infty$: the process cannot explode to infinity, it can only explode because it tends to the boundary of $E_{k_{0}}$. If moreover $k_{0}>N$ (i.e. if $\theta<2$), this is sufficient to show that $\zeta=\infty$, since then $E_{k_{0}}=({\mathbb{R}}^{2})^{N}$.

Consider a partition $K_{1},\dots,K_{p}$ of $[\![1,N]\!]$. If we neglect interactions between particles of which the indexes are not in the same subset, we have, for each $\ell\in[\![1,p]\!]$, setting ${\tilde{\theta}}_{\ell}=\theta|K_{\ell}|/N$,

and we recognize a $KS({\tilde{\theta}}_{\ell},|K_{\ell}|)$-process.

During time intervals where particles indexed in different subsets are far enough from each other, we can indeed bound the interaction between those particles, so that the Girsanov theorem tells us that $(X^{i}_{t})_{i\in K_{1}},\dots,(X^{i}_{t})_{i\in K_{p}}$ behave similarly, in the sense of trajectories, as independent $KS({\tilde{\theta}}_{1},|K_{1}|)$, …, $KS({\tilde{\theta}}_{p},|K_{p}|)$-processes.

Consider $K\subset[\![1,N]\!]$. As seen just above, during time intervals where the particles indexed in $K$ are far from all the other ones, the system $(X^{i}_{t})_{i\in K}$ behaves, in the sense of trajectories, like a $KS(\theta|K|/N,|K|)$-process. Hence, as seen in Subsection 3.2, $S_{K}(X_{t})$ behaves like a $2$-dimensional Brownian motion with diffusion constant $|K|^{-1/2}$ and $R_{K}(X_{t})$ behaves like a squared Bessel process of dimension $d_{\theta|K|/N,|K|}(|K|)$, which equals $d_{\theta,N}(|K|)$, recall (2).

Here we assume that $N>\theta\geq 2$, so that $k_{0}\in[\![2,N]\!]$ and we explain why a.s., $\zeta<\infty$ and $X_{\zeta-}=\lim_{t\to\zeta-}X_{t}$ exists, in the usual sense of $({\mathbb{R}}^{2})^{N}$.

(a) We first show that $\zeta<\infty$ a.s. On the event where $\zeta=\infty$, the squared Bessel process $R_{[\![1,N]\!]}(X)$ is defined for all times. Recall that $d_{\theta,N}(N)\leq 0$ (because $\theta\geq 2$) and that a squared Bessel process with negative dimension can be defined on the whole time half-line and a.s. becomes negative in finite time. Since $R_{[\![1,N]\!]}(X)\geq 0$ by definition, this contradicts the fact that $\zeta=\infty$. Similarly, one can show that a $KS(\theta,N)$-process has no chance to be defined after the first hitting time $\tau_{K}$ of $0$ by $R_{K}(X_{t})$, where $|K|=k_{0}$: this makes the choice of the space $E_{k_{0}}$ very natural. Indeed, assume that $X$ is defined during $[0,\zeta^{\prime})$ with $\zeta^{\prime}>\tau_{K}$. Consider the maximal subset $L$ of $[\![1,N]\!]$ containing $K$ and such that $R_{L}(X_{\tau_{K}})=0$. Then there is $\varepsilon>0$ such that during $[\tau_{K},\tau_{K}+\varepsilon]\subset[0,\zeta^{\prime})$, the particles labeled in $L$ are far from the ones labeled outside $L$. By Subsection 3.4, $(R_{L}(X_{\tau_{K}+t}))_{t\in[0,\varepsilon]}$ behaves like a squared Bessel process with dimension $d_{\theta,N}(|L|)$ issued from $0$. But such a process is instantaneously negative, because $d_{\theta,N}(|L|)\leq 0$ (since $|L|\geq k_{0}$). Since $R_{L}(X)\geq 0$, this contradicts the fact that $\tau_{K}\in[0,\zeta^{\prime})$.